Finite Group Databasebeta

← #11748752

Subgroups — #11748752

Order 960.

372 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6712C211yesmaximal, characteristic, complemented
2C2×C3×#685C211yesmaximal, characteristic
3C2×#6669C211yesmaximal, characteristic, complemented
4C2×#6668C211yesmaximal, characteristic, complemented
5C2×#6665C211yesmaximal, characteristic
6C2×#6665C211yesmaximal, characteristic
7C5×C3⋊#95C211yesmaximal, characteristic, complemented
8#64965—31nomaximal, Hall
9C23×C3×D5C2211yescharacteristic
10C22×C3×C4×C5C2211yescharacteristic
11C22×Dic15C2211yescharacteristic
12C22×Dic15C2211yescharacteristic
13C22×C5×Dic3C2211yescharacteristic
14C22×C5×Dic3C2211yescharacteristic
15C22×C3×Dic5C2211yescharacteristic
16#6712—22no—
17C3×#685—22no—
18#6669—22no—
19#6668—22no—
20#6665—22no—
21#6665—22no—
22#4948—51nomaximal, Hall
23C2×#685—31no—
24C2×#685—31no—
25C2×#685D311yescharacteristic
26C2×#683—31no—
27C2×#683—31no—
28C2×#683—31no—
29C5×#95—31no—
30C23×C3×C5C2311yescharacteristic
31C22×C3×D5—21no—
32C22×C3×D5—21no—
33C22×C3×D5—21no—
34C22×C3×D5—21no—
35C22×C3×D5—21no—
36C22×C3×D5—21no—
37C22×C3×D5Q812yes—
38C2×C3×C4×C5—21no—
39C2×C3×C4×C5—21no—
40C2×C3×C4×C5—21no—
41C2×Dic15—21no—
42C2×Dic15—21no—
43C2×Dic15—21no—
44C2×Dic15—21no—
45C2×Dic15D412yes—
46C2×Dic15D412yes—
47C2×C5×Dic3—21no—
48C2×C5×Dic3—21no—
49C2×C5×Dic3—21no—
50C2×C5×Dic3—21no—
51C2×C5×Dic3—21no—
52C2×C5×Dic3—21no—
53C2×C3×Dic5—21no—
54C2×C3×Dic5—21no—
55C2×C3×Dic5D412yes—
56C2×C3×C22⋊C4—51no—
57C2×#217—51no—
58C2×#217—51no—
59C2×C4⋊Dic3—51no—
60C2×Dic3⋊C4—51no—
61C2×Dic3⋊C4—51no—
62C3⋊#95D511yescharacteristic, complemented
63C23×D5C2×D311yescharacteristic
64C22×C4×C5—31no—
65C22×C4×C5—31no—
66C22×C4×C5C2×D311yescharacteristic
67C22×Dic5—31no—
68C22×Dic5—31no—
69C22×Dic5C2×D311yescharacteristic
70#685—22no—
71#685—62no—
72#685—62no—
73#683—62no—
74#683—62no—
75#683—62no—
76#396—151noSylow
77C22×C3×C5C2×D411yescharacteristic
78C22×C3×C5C2×D411yescharacteristic
79C22×C3×C5C2×D411yescharacteristic
80C22×C3×C5C2×Q811yescharacteristic
81C22×C3×C5#4111yescharacteristic
82C22×C3×C5#4111yescharacteristic
83C22×C3×C5#4111yescharacteristic
84C2×C3×D5—22no—
85C2×C3×D5—22no—
86C2×C3×D5—22no—
87C2×C3×D5—41no—
88C2×C3×D5—41no—
89C2×C3×D5—41no—
90C2×C3×D5—41no—
91C3×C4×C5—41no—
92Dic15—22no—
93Dic15—22no—
94C3×Dic5—22no—
95C5×Dic3—41no—
96C5×Dic3—41no—
97C24×C3—51no—
98C22×C3×C4—51no—
99C22×C3×C4C2×D511yescharacteristic
100C22×Dic3—51no—
101C22×Dic3—51no—
102C22×Dic3C2×D511yescharacteristic
103C22×Dic3C2×D511yescharacteristic
104C3×C22⋊C4—102no—
105#217—102no—
106#217—102no—
107C4⋊Dic3—102no—
108Dic3⋊C4—102no—
109Dic3⋊C4—102no—
110C23×C5C22×D311yescharacteristic
111C22×D5—21no—
112C22×D5—21no—
113C22×D5—21no—
114C22×D5—21no—
115C22×D5—21no—
116C22×D5—21no—
117C22×D5Dic612yes—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—61no—
122C2×C4×C5—61no—
123C2×C4×C5—61no—
124C2×C4×C5—61no—
125C2×C4×C5—61no—
126C2×C4×C5—61no—
127C2×Dic5—21no—
128C2×Dic5—21no—
129C2×Dic5—32no—
130C2×Dic5—32no—
131C2×Dic5—61no—
132C2×Dic5—61no—
133C2×Dic5—61no—
134C2×Dic5—61no—
135C2×Dic5#6712yes—
136C2×C4⋊C4—151no—
137C2×C4⋊C4—151no—
138C2×C4⋊C4—151no—
139C2×C22⋊C4—51no—
140C2×C22⋊C4—151no—
141C2×C22⋊C4—151no—
142#95—31no—
143C2×C3×C5#12111yescharacteristic
144C2×C3×C5#12111yescharacteristic
145C2×C3×C5#12111yescharacteristic
146C2×C3×C5C22⋊Q811yescharacteristic
147C2×C3×C5C22⋊Q811yescharacteristic
148C2×C3×C5C22⋊Q811yescharacteristic
149C2×C3×C5#12611yescharacteristic
150C3×D5—42no—
151C23×C3—52no—
152C23×C3—101no—
153C23×C3—101no—
154C23×C3—101no—
155C23×C3—101no—
156C23×C3—101no—
157C23×C3—101no—
158C23×C3C22×D511yescharacteristic
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—52no—
163C2×C3×C4—101no—
164C2×C3×C4—101no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—21no—
168C2×Dic3—21no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—52no—
172C2×Dic3—52no—
173C2×Dic3—101no—
174C2×Dic3—101no—
175C2×Dic3—101no—
176C2×Dic3—101no—
177C22×C5C2×Dic611yescharacteristic
178C22×C5#23511yescharacteristic
179C22×C5D3×D411yescharacteristic
180C22×C5D3×D411yescharacteristic
181C22×C5#23711yescharacteristic
182C22×C5#23711yescharacteristic
183C22×C5C2×#6711yescharacteristic
184C2×D5—22no—
185C2×D5—22no—
186C2×D5—22no—
187C2×D5—41no—
188C2×D5—41no—
189C2×D5—41no—
190C2×D5—41no—
191C4×C5—41no—
192C4×C5—121no—
193C4×C5—121no—
194Dic5—22no—
195Dic5—62no—
196Dic5—62no—
197C24—51no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—51no—
201C22×C4—151no—
202C22×C4—151no—
203C22×C4D3×D511yescharacteristic
204C4⋊C4—302no—
205C4⋊C4—302no—
206C4⋊C4—302no—
207C22⋊C4—102no—
208C22⋊C4—302no—
209C22⋊C4—302no—
210C3×C5#39611yescharacteristic, complemented, Hall
211C22×C3—102no—
212C22×C3—102no—
213C22×C3—102no—
214C22×C3—201no—
215C22×C3—201no—
216C22×C3—201no—
217C22×C3—201no—
218C22×C3#70911yescharacteristic
219C22×C3#70911yescharacteristic
220C22×C3#70911yescharacteristic
221C22×C3D4×D511yescharacteristic
222C22×C3D4×D511yescharacteristic
223C22×C3D4×D511yescharacteristic
224C22×C3Q8×D511yescharacteristic
225C3×C4—41no—
226C3×C4—102no—
227Dic3—41no—
228Dic3—41no—
229Dic3—102no—
230Dic3—102no—
231C2×C5#87811yescharacteristic
232C2×C5#87811yescharacteristic
233C2×C5#87911yescharacteristic
234C2×C5#88411yescharacteristic
235C2×C5#92411yescharacteristic
236C2×C5#93811yescharacteristic
237C2×C5#93911yescharacteristic
238D5—42no—
239C23—52no—
240C23—101no—
241C23—101no—
242C23—101no—
243C23—101no—
244C23—101no—
245C23—101no—
246C23C2×D3×D511yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—52no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—101no—
258C2×C4—101no—
259C2×C4—152no—
260C2×C4—152no—
261C2×C4—301no—
262C2×C4—301no—
263C2×C4—301no—
264C2×C4—301no—
265C2×C3—202no—
266C2×C3#402211yescharacteristic
267C2×C3#403011yescharacteristic
268C2×C3#403011yescharacteristic
269C2×C3#403011yescharacteristic
270C2×C3#404211yescharacteristic
271C2×C3#404211yescharacteristic
272C2×C3#404211yescharacteristic
273C5#494811yescharacteristic, complemented, Sylow
274C22—102no—
275C22—102no—
276C22—102no—
277C22—201no—
278C22—201no—
279C22—201no—
280C22—201no—
281C22D5×Dic611yescharacteristic
282C22#676511yescharacteristic
283C22#677111yescharacteristic
284C22#677711yescharacteristic
285C22#677911yescharacteristic
286C22D5×#6711yescharacteristic
287C22#679011yescharacteristic
288C4—41no—
289C4—102no—
290C4—121no—
291C4—121no—
292C4—302no—
293C4—302no—
294C3#6496511yescharacteristic, complemented, Sylow
295C2—202no—
296C2#9134911yescharacteristic
297C2#9136311yescharacteristic
298C2#9140811yescharacteristic
299C2#9143011yescharacteristic
300C2#9143111yescharacteristic
301C2#9146111yescharacteristic
302C2#9153311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D5
commutator, socle, cosocleC23×C3×C5C23
FittingC22×C3×C4×C5C22
prime radicalC23×C3×D5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#396
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#6712C2semidirect
3. C2×#6669C2semidirect
4. C2×#6668C2semidirect
7. C5×C3⋊#95C2semidirect
62. C3⋊#95D5semidirect
210. C3×C5#396semidirect
273. C5#4948semidirect
294. C3#64965semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#67122
C2×C3×#6852
C2×#66692
C2×#66682
C2×#66652
C5×C3⋊#952
#649653
#49485